An Investigation on Left Hyperideals of Ordered Semihypergroups
Received:January 20, 2016  Revised:May 24, 2017
Key Word: ordered semihypergroup   minimal left hyperideal   maximal left hyperideal   weakly prime left hyperideal   quasi-prime left hyperideal   weakly quasi-prime left hyperideal  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11361027), the University Natural Science Project of Anhui Province (Grant No.KJ2015A161), the Natural Science Foundation of Guangdong Province (Grant No.2014A030313625) and the Key Project of Department of Education of Guangdong Province (Grant No.2014KZDXM055).
Author NameAffiliation
Jian TANG School of Mathematics and Statistics, Fuyang Normal University, Anhui 236037, P. R. China 
Xiangyun XIE School of Mathematics and Computational Science, Wuyi University, Guangdong 529020, P. R. China 
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      In this paper, the concepts of minimal and maximal left hyperideals in ordered semihypergroups are introduced, and several related properties are investigated. Furthermore, we introduce the concepts of weakly prime, quasi-prime, quasi-semiprime and weakly quasi-prime left hyperideals of an ordered semihypergroup, and establish the relationship among the four classes of left hyperideals. Moreover, we give some characterizations of weakly quasi-prime left hyperideals by the left hyperideals and weakly $m$-systems. We also characterize the quasi-prime left hyperideals in terms of the $m$-systems. In particular, we prove that an ordered semihypergroup $S$ is strongly semisimple if and only if every left hyperideal of $S$ is the intersection of all quasi-prime left hyperideals of $S$ containing it.
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